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  #1  
Old 02-24-2005, 08:55 AM
Khadaji Khadaji is offline
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Which would you prefer: A half a barrel of dimes or a barrel of nickels?

Is there a way to mathmatically determine the answer to the question: Which would you prefer: A half a barrel of dimes or a barrel of nickels?

It seems to me that there isn't enough information. But I haven't had any mathmatics in many years.
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  #2  
Old 02-24-2005, 09:02 AM
Fuji Kitakyusho Fuji Kitakyusho is offline
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Stacking geometry notwithstanding, you can answer this problem on the basis of the relative volume of the two coins. Assuming that the barrel referred to in each case is of the same volume, look a a dime, and then a nickel - you will notice that the volume of a dime is considerably less than half of the volume of a nickel. When you fill a barrel half full of dimes, it stands to reason that the value of these coins will be greater than a full barrel full of nickels.
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  #3  
Old 02-24-2005, 09:10 AM
erislover erislover is offline
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Dimes and quarters, on the other hand...
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  #4  
Old 02-24-2005, 09:12 AM
Uncommon Sense Uncommon Sense is offline
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I'll take a half-barrel that is half-full half-way right on up to the half-way point with dimes, but only if half of it's filled.
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  #5  
Old 02-24-2005, 09:14 AM
galen galen is offline
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Dimes, I think.
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  #6  
Old 02-24-2005, 09:17 AM
Tranquilis Tranquilis is offline
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Dimes, most obviously - Not only can you get at least twice as many in the same volume, they're twice as valuable coin-for-coin, meaning that the barrel of dimes would be at least four times more valuable than the nickels.
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  #7  
Old 02-24-2005, 09:18 AM
John T. Conklin John T. Conklin is offline
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Quote:
Originally Posted by Khadaji
Is there a way to mathmatically determine the answer to the question: Which would you prefer: A half a barrel of dimes or a barrel of nickels?

It seems to me that there isn't enough information. But I haven't had any mathmatics in many years.
I had no luck googling for the volume of a dime or nickel, but I since I happened to have my trusty dial calipers sitting on my desk, I was able to measure the diameter and thickness of a "representitive" dime and nickel I had in my change bucket.

The dime measured 0.796" in diameter and 0.050" thick. The nickel measured 0.838" in diameter and 0.075" thick.

Since cylinders don't pack well, I calculated volume as if the surface was a square rather than a circle. (This is where I may be off, is the packing better if they're arranged in "hexagons"). The dime's "volume" is 0.0316 in^3 and the nickel's "volume" is 0.0527 in^3.

Since the dime's volume is greater than half the nickel's volume, but it is only twice the value, I think the better choice is the full barrel of nickels.
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  #8  
Old 02-24-2005, 09:24 AM
John T. Conklin John T. Conklin is offline
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Interesting that everyone else has chosen dimes while I've been off measuring...

It seems that relative volumes on this scale are difficult to estimate.
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  #9  
Old 02-24-2005, 09:27 AM
Munch Munch is offline
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Quote:
Originally Posted by Tranquilis
Dimes, most obviously - Not only can you get at least twice as many in the same volume, they're twice as valuable coin-for-coin, meaning that the barrel of dimes would be at least four times more valuable than the nickels.
Read the OP again. You pick either:

1. A Full barrel of nickels

-or-

2. A HALF barrel of dimes.

I'd still go with the dimes.
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  #10  
Old 02-24-2005, 09:27 AM
chrisk chrisk is online now
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Quote:
Originally Posted by John T. Conklin
Since the dime's volume is greater than half the nickel's volume, but it is only twice the value, I think the better choice is the full barrel of nickels.
Explain that please?? Even if they have the same volume, as long as the dime is twice the value of the nickel the barrels would be of equivalent value. ANY space savings with the dime translates into an advantage.

Of course, that's only if you're interested in maximizing cash value. If your personal preferences depend on other criteria, that's your decision.
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  #11  
Old 02-24-2005, 09:35 AM
Chefguy Chefguy is online now
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Well, if this was in the mix, I'd take the nickels.
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  #12  
Old 02-24-2005, 09:40 AM
Joey P Joey P is offline
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This reminds me of a joke I just read:

There's a little fellow named Junior who hangs out at the local grocery store. The manager doesn't know what Junior's problem is, but the boys like to tease him. They say he is two bricks short of a load, or his elevator doesn't go all the way to the top.

To prove it, sometimes the boys offer Junior his choice between a nickel and a dime. He always takes the nickel, they say, because it's bigger.

One day after Junior grabbed the nickel, the store manager got him off to one side and said, "Junior, those boys are making fun of you. They think you don't know the dime is worth more than the nickel. Are you grabbing the nickel because it's bigger, or what?"

Junior said, "Well, if I took the dime, they'd quit doing it!"
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  #13  
Old 02-24-2005, 09:41 AM
zut zut is offline
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Quote:
Originally Posted by John T. Conklin
I had no luck googling for the volume of a dime or nickel, but I since I happened to have my trusty dial calipers sitting on my desk, I was able to measure the diameter and thickness of a "representitive" dime and nickel I had in my change bucket.

The dime measured 0.796" in diameter and 0.050" thick. The nickel measured 0.838" in diameter and 0.075" thick.
You sure you measured right? The US Mint says the diameter of a dime is about 0.705 in. Moreover, this page claims that there are about 5250 nickels per gallon, and about 10500 dimes, for an exact 2:1 ratio. [And, not on topic, but interesting: dimes, quarters, and half dollars work out to the same amount of money per gallon.]
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  #14  
Old 02-24-2005, 09:46 AM
Monstre Monstre is offline
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Quote:
Originally Posted by chrisk
Explain that please?? Even if they have the same volume, as long as the dime is twice the value of the nickel the barrels would be of equivalent value. ANY space savings with the dime translates into an advantage.
Exactly. Since a dime is not larger than a nickel, I can't see any way that a full barrel of nickels would ever have more monetary value than the half-barrel of dimes (assuming we're talking the same size barrel here).

At worst, you'd have half as many dimes as nickels (same monetary value), and more likely you'd have at least some extra dimes.

And the half-barrel of dimes would be lighter -- easier to carry to the bank.

Now, if you'll offer me a half-barrel of paper money, I'll forget I ever heard about the dimes.
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  #15  
Old 02-24-2005, 10:14 AM
chrisk chrisk is online now
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Quote:
Originally Posted by Monstre
Now, if you'll offer me a half-barrel of paper money, I'll forget I ever heard about the dimes.
How about a briefcase?
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  #16  
Old 02-24-2005, 10:28 AM
Kimstu Kimstu is offline
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JTC: Since cylinders don't pack well, I calculated volume as if the surface was a square rather than a circle. (This is where I may be off, is the packing better if they're arranged in "hexagons").

Yes, it is. The hexagonal lattice packing (with each circle surrounded by six circles touching it) is the maximally dense packing of uniformly-sized circles.

(Math Phun Phact: The corresponding statement for three dimensions, that hexagonal lattice packing of uniformly-sized spheres is maximally dense, was conjectured by Johannes Kepler back in 1611 but not actually proved until a few years ago, by Thomas Hales.

Physics Phun Phact: some dude whose name I don't remember experimentally tested the Kepler conjecture by leaving dried peas in a pot of water overnight and noting that their expansion had squeezed them into little dodecahedrons, suggesting that the space-saving distribution they naturally settled into was the hexagonal lattice packing. This illustrates why the Kepler conjecture always used to be introduced with the words "Many mathematicians believe, and all physicists know, that..."

I always get a chuckle out of thinking of that guy counting the little faces on a soggy pea. )
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  #17  
Old 02-24-2005, 10:40 AM
ultrafilter ultrafilter is offline
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Weigh the barrels. Let w1 be the weight of the barrel of nickels, and w2 be the weight of the barrel of dimes. Let d be the weight of a dime, and n the weigh of a nickel.

The value of the barrel of dimes is w1/10d, and the value of the barrel of nickels is w2/5n. If w1/w2 > 2d/n, then the barrel of dimes is worth more. Otherwise, it's the barrel of nickels.
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  #18  
Old 02-24-2005, 10:56 AM
chrisk chrisk is online now
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Quote:
Originally Posted by Kimstu

(Math Phun Phact: The corresponding statement for three dimensions, that hexagonal lattice packing of uniformly-sized spheres is maximally dense, was conjectured by Johannes Kepler back in 1611 but not actually proved until a few years ago, by Thomas Hales.

Physics Phun Phact: some dude whose name I don't remember experimentally tested the Kepler conjecture by leaving dried peas in a pot of water overnight and noting that their expansion had squeezed them into little dodecahedrons, suggesting that the space-saving distribution they naturally settled into was the hexagonal lattice packing. This illustrates why the Kepler conjecture always used to be introduced with the words "Many mathematicians believe, and all physicists know, that..."

I always get a chuckle out of thinking of that guy counting the little faces on a soggy pea. )
What exactly is the connection between hexagonal lattices for spheres, and dodecahedrons... I don't believe that dodecahedrons will pack together in a hexagonal lattice... or any lattice very neatly. I could be wrong.
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Old 02-24-2005, 10:58 AM
John Corrado John Corrado is offline
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It's not mathematics, it's just simple logic.

Assume that dimes and nickels are exactly the same shape.

Fill half a barrel with dimes, and fill a barrel with nickels. By definition, the two are worth the exact same value- the barrel of nickels has twice as many coins, but each coin is worth half the value of a dime.

But, because nickels are larger than dimes, there must be *more* dimes in the half-barrel than half the number of nickels in the full barrel. Therefore, the half-barrel of dimes is worth more.
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  #20  
Old 02-24-2005, 11:03 AM
Kimstu Kimstu is offline
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chrisk: I don't believe that dodecahedrons will pack together in a hexagonal lattice... or any lattice very neatly. I could be wrong.

I don't think you're wrong about regular dodecahedra, but the ones that hexagonal-lattice-packed spheres expand into are rhombic dodecahedra, which are space-filling:
Quote:
If spheres packed in a cubic lattice, face-centered cubic lattice, and hexagonal lattice are allowed to expand, they form cubes, hexagonal prisms, and rhombic dodecahedra.
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  #21  
Old 02-24-2005, 11:05 AM
John T. Conklin John T. Conklin is offline
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Quote:
Originally Posted by zut
You sure you measured right? The US Mint says the diameter of a dime is about 0.705 in.
I measured again, and got 0.703".

I'm not quite sure how I managed to be off by .090" Might as well be a mile.

Quote:
Moreover, this page claims that there are about 5250 nickels per gallon, and about 10500 dimes, for an exact 2:1 ratio. [And, not on topic, but interesting: dimes, quarters, and half dollars work out to the same amount of money per gallon.]
With the new numbers, the cubic volume of the dime is 0.0248 in^3, very close to half the volume of the nickel. Taking the next step, Dimes are ~403 cents/in^3 and nickels are ~95 cents/in^3. So it looks like a half barrel of dimes is still about twice is valuable as a full barrel of nickels.

So I guess I should have taken the half barel after all. I guess that's the difference between precision and accuracy...
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  #22  
Old 02-24-2005, 11:08 AM
chrisk chrisk is online now
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Quote:
Originally Posted by John Corrado
It's not mathematics, it's just simple logic.

**snip**

But, because nickels are larger than dimes, there must be *more* dimes in the half-barrel than half the number of nickels in the full barrel. Therefore, the half-barrel of dimes is worth more.
The emphasized sentence is a weak logical step, it seems to me. Depending on the size and shape of the dime, it's not guaranteed that there are more dimes than half the full barrel of nickels... if the dimes are a little smaller and much more awkwardly shaped, there might not even be as many.

Given that dimes and nickels are the same overall shape (highly flattened cylinders,) I think you can make a convincing argument that there are at least as many dimes as half the nickels, and therefore that the dimes are at least as valuable. To prove anything further, you kinda need to go to the maths.

Quote:
Originally Posted by Kimstu
I don't think you're wrong about regular dodecahedra, but the ones that hexagonal-lattice-packed spheres expand into are rhombic dodecahedra, which are space-filling:
Oooh, interesting. I tend to associate those terms, (dodecahedron, tetrahedron, icosahedron, octahedron) so strongly with the platonic solids that I never even thought about irregular solids.
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  #23  
Old 02-24-2005, 11:10 AM
John Corrado John Corrado is offline
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Actually, to express it mathematically:

Let VB= The volume of the Barrel.
Let VD= The volume of a dime.
Let VN= The volume of a nickel.

The question is the relationship between

1/2*(VB/VD)*10 and (VB/VN)*5.

which can be reduced to

(VB/VD)*5 vs. (VB/VN)*5

which goes down to

(VB/VD) vs. (VB/VN)

which can further be reduced to

(1/VD) vs. (1/VN).

Since VD<VN, 1/VD>1/VN.
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  #24  
Old 02-24-2005, 11:35 AM
Tranquilis Tranquilis is offline
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Quote:
Originally Posted by Munch
Read the OP again. You pick either:

1. A Full barrel of nickels

-or-

2. A HALF barrel of dimes.

I'd still go with the dimes.
Oops. Quite right.

I'd still take dimes - I'm betting they're less than half the volume of a nickel, and even if they aren't, they're still easier to move about.
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Old 02-24-2005, 12:05 PM
HeyHomie HeyHomie is offline
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Quote:
Originally Posted by zut
Moreover, this page claims that there are about 5250 nickels per gallon, and about 10500 dimes, for an exact 2:1 ratio.
A 30-gallon barrell of nickels: 157,500 nickels = $7,875.

15 gallons of dimes: 157,500 dimes = $15,750.

Take the dimes.
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  #26  
Old 02-24-2005, 12:14 PM
Bottle of Smoke Bottle of Smoke is offline
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I'm with John on this. Unless I'm totally missing something. Which is certainly possible. I don't know how to write a math formula for it, but logic says the dimnes win.

Pretend that nickles and dimes are exactly the same size coin.
Pretend that a full barrel hold 1,000 coins (that's one small barrel, but bear with me).

A full barrel of nickels would be 1,000 coins at $0.05 each, or $50.
A half barrel of dimes would be 500 coins at $0.10 each, or $50.

So only in the case of the two coins being the same size would the value of both barrels be the same.

But dimes are clearly much smaller than nickles, so a half barrel of dimes would be much more than 500 coins. Which means that the half barrel would have a much higher value.

(If I missed something, go easy on me. I ain't had no fancy book larnin')
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  #27  
Old 02-24-2005, 02:26 PM
Enright3 Enright3 is online now
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Quote:
Originally Posted by chrisk
How about a briefcase?
Sorry, can't do that... only Marsellus Wallace knows what's in the briefcase.
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  #28  
Old 02-24-2005, 03:19 PM
Uncommon Sense Uncommon Sense is offline
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Originally Posted by Bottle of Smoke
I'm with John on this. Unless I'm totally missing something. Which is certainly possible. I don't know how to write a math formula for it, but logic says the dimnes win.

Pretend that nickles and dimes are exactly the same size coin.
Pretend that a full barrel hold 1,000 coins (that's one small barrel, but bear with me).

A full barrel of nickels would be 1,000 coins at $0.05 each, or $50.
A half barrel of dimes would be 500 coins at $0.10 each, or $50.

So only in the case of the two coins being the same size would the value of both barrels be the same.

But dimes are clearly much smaller than nickles, so a half barrel of dimes would be much more than 500 coins. Which means that the half barrel would have a much higher value.

(If I missed something, go easy on me. I ain't had no fancy book larnin')
I agree. They're twice the value at half the size. Dimes win.
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  #29  
Old 02-24-2005, 03:42 PM
chrisk chrisk is online now
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Quote:
Originally Posted by Enright3
Sorry, can't do that... only Marsellus Wallace knows what's in the briefcase.
Not *that* briefcase. Just an ordinary briefcase full of... well, let's say one dollar bills. Make your pick between that and the other two choices.
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  #30  
Old 02-24-2005, 06:01 PM
Xema Xema is offline
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Quote:
Originally Posted by chrisk
The emphasized sentence is a weak logical step, it seems to me. Depending on the size and shape of the dime, it's not guaranteed that there are more dimes than half the full barrel of nickels... if the dimes are a little smaller and much more awkwardly shaped, there might not even be as many.
It would be quite surprising if a smaller, thinner coin somehow managed to take up more space than a larger, fatter one.
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  #31  
Old 02-24-2005, 09:23 PM
Civil Guy Civil Guy is offline
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Quote:
Originally Posted by Tranquilis
Oops. Quite right.

I'd still take dimes - I'm betting they're less than half the volume of a nickel, and even if they aren't, they're still easier to move about.
I agree. There's more chance we could move the barrel ourselves, rather than have to cough up a few bucks hiring someone to help us. 'Course, I dunno... I bet even a half barrel of dimes is still pretty heavy...
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  #32  
Old 02-24-2005, 09:31 PM
Shagnasty Shagnasty is offline
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Quote:
Originally Posted by chrisk
The emphasized sentence is a weak logical step, it seems to me. Depending on the size and shape of the dime, it's not guaranteed that there are more dimes than half the full barrel of nickels... if the dimes are a little smaller and much more awkwardly shaped, there might not even be as many.
What are all these different shapes that you have seen dimes come in?
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